Try
here:
122 W 9th 1 bedroom main floor Mitchell (http://sd.craigslist.org/apa/2752368991.html - broken link)
here:
719 S. Edmunds 2 bedroom lower level Mitchell (http://sd.craigslist.org/apa/2752367513.html - broken link)
here:
127 1/2 E 2nd #4, small efficiency in Mitchell (http://sd.craigslist.org/apa/2752361891.html - broken link)
Try here:
2 bedroom with free cable and some utilities paid (http://siouxfalls.craigslist.org/apa/2735981044.html - broken link)
Those are a start. Also try calling the local newspaper and ask them for any tips.
The Daily Republic | Mitchell, South Dakota
I know there are some in Salem which is 30 minutes from Mitchell. Actually one is very cheap, nice, all utilities and pets are allowed. If you're interested let me know, I can find the link.
Hope that helps